Physics Phase Transitions in Anisotropic Lattice Spin ... - Project Euclid

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Commun. math. Phys. 60, 233—267 (1978)

Physics ©by Springer-Verlag 1978

Phase Transitions in Anisotropic Lattice Spin Systems Jurg Frόhlich1* and Elliott H. Lieb 2 ** Department of Mathematics1 and Departments of Mathematics and Physics2, Princeton University, Princeton, New Jersey 08540, USA

Abstract. A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity is not proved, but for which the rest of the proof is valid. I. Summary of Results and General Strategy of Proofs One of the main purposes of this paper is to explain a general method for proving the existence of phase transitions, in the sense of long range order at sufficiently low temperatures, in classical and quantum lattice systems. In principle, our method can be applied to arbitrary lattice systems satisfying reflection positivity (a condition closely related to the existence of a self-adjoint positive definite transfer matrix), the groundstates of which are essentially finitely degenerate (e.g. the space of groundstates decomposes into finitely many subspaces labelled by a discrete order parameter, sometimes related to a broken discrete symmetry group). Our method is inspired by recent work of Glimm, Jaffe and Spencer concerning phase transitions in the (λφ4)2 quantum field model, [16]. In this paper their ideas are extended in two ways: 1. We systematize the use of reflection positivity and chessboard estimates * Present address: Institut des Hautes Etudes Scientifϊques, F-91440 Bures-sur-Yvette, France. A Sloan Foundation fellow. Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864 ** Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01

0010-3616/78/0060/0233/$07.00

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J. Frόhlich and E. Lieb

in obtaining upper bounds on the statistical weight of contours arising in a Peierls argument and we show how to apply these methods to quantum lattice systems. This reduces the proof of long range order to estimating a ratio between a constrained partition function and the usual partition function. (This is basically a thermodynamic estimate). 2. We introduce the principle of exponential localization in order to derive upper bounds on constrained partition functions. This principle is particularly useful in the analysis of quantum lattice systems. Reflection positivity, originally inspired by work of Osterwalder and Schrader [18], and the principle of exponential localization are useful tools in contexts other than the theory of phase transitions. In Section LA we introduce six different classical and quantum mechanical models on the two dimensional square lattice in terms of which we develop and illustrate our general method. A summary of our main results concludes that section. In Section I.B we recall the connections between phase transitions and the occurrence of various forms of long range order (LRO) at sufficiently low temperatures. In Section I.C, D and E we present the main ideas behind our general method; (Section I.C contains a convenient variant of the Peierls argument, essentially identical to the one of [16]; see also [10]). In Section II we establish, (or review) reflection positivity for five of our six models, the exception being the quantum Heisenberg ferromagnet. We prove a generalization of the Holder inequality for traces which, when combined with reflection positivity, yields the chessboard estimates. They extend constructive field theory estimates of [19]. In Section III we introduce the principle of exponential localization and apply it to our models for the purpose of estimating constrained partition functions. This is an expansion of the idea used in [22]. In Section IV the proofs of our main results are completed by combining the estimates of Sections I.C, II and III. Sections II and III contain results which are of some interest in their own right: Theorems 2.1, 2.2, 3.1 and Corollary 3.2. The reader can understand their statements and proofs without being familiar with the rest of this paper. Next, we describe the models studied in this paper in general terms and recall some typical aspects of two dimensional lattice systems. Two facts are well established about two-dimensional (quantum or classical) lattice spin systems with short range interactions: (i) The Ising model has a first order phase transition (i.e. long range order for large β = {kT)~x); for all values S = 1/2, 1,... of the spin. (ii) Models with continuous symmetry (e.g. the isotropic Heisenberg models) have no such ordering. The proof of this is due to Mermin-Wagner [1], Mermin [2] and Hohenberg [3], (MWH). Thus, a natural question is whether the anisotropic models have LRO for all values of the anisotropy parameter, α, with 0 < a < 1. For the classical Heisenberg (H) model this was proved recently by Malyshev [4]. Kunz, Pfister and Vuillermot [5] later gave a simplified proof

Phase Transitions in Anisotropic Lattice Spin Systems

235

for the planar rotator. Ginibre [6] and Robinson [7] proved LRO for the quantum Heisenberg jerromagnet for very small α. In [8] we announced proofs of LRO for a variety of anisotropic models: in particular for the quantum ferromagnetic H model, for all α < 1. Subsequently we became aware of a flaw in one of the lemmas for the ferromagnetic model. This is basically the same flaw as in the announced Dyson, Lieb and Simon [9] proof of LRO for the three dimensional H model. The other results stated in [8] are correct. Here we will present the details of the proofs, including the part of the proof for the ferromagnetic H model that is correct. It is hoped that before very long the missing piece of the puzzle will be filled in. An obvious remark has to be made: All the models we consider have no LRO at high temperature, a fact which can be proved by high temperature expansions, for example. Since LRO implies the existence of a spontaneously polarized state, our proof of LRO at low temperature implies the existence of a phase transition. The models discussed here are all two-dimensional but, as in the usual Peierls argument [23], all our results and methods can be extended to higher dimensions. They can also be extended to some other lattices, e.g. the honeycomb lattice; see [12]. Some of our results were reviewed in [10] and [11]. Additional applications of the ideas presented here are to be found [10], [12] and [13]. LA. Description of the Models and Main Results All models are on the square lattice Z 2 and have only nearest neighbor interactions. Thus ]Γ means a sum over nearest neighbors, each term being included once; H is the Hamiltonian. (1) Classical N-Vector Model (N > 1) )\

(1.1)

N

Each S. is a unit vector in U , uniformly distributed on the sphere. (More general rotationally symmetric spin distributions could be accomodated by our methods.) Note that in this classical case, the ferromagnet (minus sign in (1.1)) is equivalent to the antiferromagnet (plus sign) by reversing the spins on the odd sublattice. This is not true in quantum models. Our result in this case is that for every a < 1 there is LRO at low T. In other words for every α < 1 there is a βc{a) such that there is LRO for β > βc(a). Our estimate on βc(a) is βc(a) = O((l-a)~1).

(1.2)

The MWH result is that βc(0, 0 with ih

Therefore

0 and β large enough

uniformly in A and j . Thus, II > - ε .

(1.15)

Next we discuss term III on the right side of (1.11). By the Schwarz inequality for the state < — >yl Λ^Λ,

(i.i6)

and we have used *Σ

Σ ( Π PΪP

1 = 2 {γ:\y\ = 2l} \ ey

(When A is finite these sums are finite). Given some fixed length 2/, well known combinatorics shows that there are no more than 2(1 — 1)32Z~2 contours of length 2/, provided A is large enough, depending on m and n. (The factor 3 2 *" 2 comes from a standard 1— -argument and the fact that all contours consist of one or two closed pieces. The factor 2(1 — 2) comes from the fact that each contour must separate m from ή). Theorem 1.1 now follows from (1.33) and the inequality 00

2l

2

2lκ

K > In 3 which guarantees that the series Σ 2(/ — l)3 ~ e~ '=* Theorem 1.1 has the following

converges. Q.E.D.

Corollary 1.2. Given ε > 0, there exists some finite K(ε) such that, for all K ^ K(s\

Remarks. 1. The relevance of Corollary 1.2 for the proof of long range order has been explained in Section l.B. 2. Theorem 1.1 and Corollary 1.2 can easily be generalized to the case of more than two positive operators (e.g. projections) Pj,..., Pjf say, with M 1=1

Phase Transitions in Anisotropic Lattice Spin Systems

243

We could apply this more refined Peierls argument to the quantum ferromagnet, model (4), with Pj = P.+*, Pf = P., we obtain

Π PΪPJ ) £ Π < Π

PΪPJ

.

(1.43)

To each term on the r.s. of (1.43) we apply the Schwarz inequality (1.40) repeatedly, for all allowed choices of reflections θt at lattice lines I This yields (see Section II)

Π j>ey*,β

Kp7 ) = 2 | y " ' l/μi1 .

(1.44)

'

The key inequality (1.38) follows from inequalities (1.43) and (1.44). Further details concerning reflection positivity and the chessboard estimates (1.42) and (1.44) are given in Section II. I.E. Estimate o/< PΛ > and Exponential Localization In this section we sketch the main ideas of how to estimate

By definition of < — >, • Tr(exp[-/?tfJ)

*

where R(AC) means either RΛ or RAC. Here HΛ is the Hamiltonian of the model under consideration, and Tr is the usual trace in the quantum mechanical models, and, in the classical models, an integral with measure the product of the single spin distributions over all sites in A. Let EΛ(de) denote the spectral measure of the Hamiltonian HΛ. By the spectral theorem Tr(exp [ - βHjC) = ]e~βe Ύr(EΛ(de)C),

(1.46)

eo

where e0 = eo{A) = infspec HΛ is the groundstate energy, and C is an arbitrary operator, resp. function. We will choose some positive number Δ = Δ(β), depending c on the model under consideration, and decompose R^ \β) into two pieces eo + Δ\Λ\

de)PJAC)),

]

Λ

(1.47)

eo + Λ\Λ\

where = f e-βeΎr(EΛ(de))

(1.48)

Phase Transitions in Anisotropic Lattice Spin Systems {

247

C)

is the partition function. We estimate R + (β,A) by 1

exp { - β[e0 + A \Λ | ] }

β, A) S ZΛ(β)~

J

Ύv{EΛ{de)) 1

S exp{ - /?[e0 + A \Λ{]} { T r ( l ) Z ^ ) - } .

(1.49)

The Peierls-Bogoliubov inequality will be shown to give Ύr(l)/ZΛ(β) ^ exp β[e0 + ^ | Λ | ] ,

(1.50)

for β sufficiently large. Thus R^{β9A) S exp{ - βΔ\Λ\/2}. {

(1.51)

c

Next we consider R ^ \β, A). In the classical cases this will vanish for the following reason: A will be chosen sufficiently small so that P{£C) will be a projection onto configurations with energy greater than e o + J | y l | . Thus the integral for R^c\β) will vanish identically. In the quantum cases, the situation is more complicated. Although PΛ will be a projection onto states whose average energy exceeds e0 + A \Λ\, the integral does not vanish, because PΛ will have nonvanishing matrix elements in eigenstates of HΛ with energy < e0 + A \A\. To be explicit, let e0 ^ eγ ^ be the eigenvalues of HΛ with eigenvectors φ0, φ1,.... Then R_(β, A) = ZΛ(β)~1Σ'C;

exp[ - βej

(1.52)

where £ ' means a sum over i such that e. ^ e0 + zl | /I |, and C^iφ^P^).

(1.53)

Now C is independent of jβ, and terefore R_(β,A) does not necessarily vanish as exp[ — β(const.)]. What we have to show is that C. -> 0 sufficiently fast as i -> 0. Then we can hope that R_(β,A) goes to zero sufficiently fast as β -• oo, for a suitable choice of Zl. The estimate on C , carried out in Section III, comes about in the following way: We write HΛ = AΛ + BΛ, where BΛ is suitably small compared to AA9 and such that PΛ is an eigenprojection for AΛ onto AΛ eigenvectors having energy > e0 + nΔ \A I for some n. In model (3), for example, A = S~2HZ. If BΛ were zero then C. = 0 for e. ^ e 0 + J |yl|. The principle of exponential localization will tell us that ^ eigenvectors of A^ energy greater than eo +nA\A\,(n^2) when expanded in the φt, are strongly (indeed, exponentially well in |vl|) localized around φ. with β. > e0 + zl | A |. This, in turn, will lead to C. being smα// for e^ e0. Acknowledgements. We thank B. Simon for some very useful suggestions.

II. Reflection Positivity and Chessboard Estimates II.A. Reflection Positivity In this section we recall the proofs of reflection positivity, inequality (1.39), for the models studied in this paper. For the classical iV-vector models, reflection

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J. Frόhlich and E. Lieb

positivity is shown in [20]. In terms of a transfer matrix formalism it is used in [10]. For the quantum anti-ferromagnet and the quantum mechanical xy model (models (3) and (6)) reflection positivity was discovered in [9]. The proof given there also applies to the classical JV-vector models. First we consider the classical, anharmonic crystal, model (2), for which (1.39) is new. We choose a pair of lattice lines / cutting A into two congruent pieces, Λ+ and Λ_, with Λ+ nΛ_ = I Let Λ± = Λ±\l The N-vector oscillators attached to sites in Λ± have coordinates (y)± = tyeR1* :jeA±}. The coordinates of the JV-vector oscillators attached to sites in / are denoted by Given a function F of (y)+, (z), we define ΘJF to be the function of (y)_, (z) obtained by substituting yθιj for yj9 for a\\jeA±, i.e. Of is the reflection of F in the lines /. The Hamilton function HΛ of the AC model is given by HΛ=

= Σ = Σ Let dx be the a priori distribution of a single oscillator, and set

Let F = F((y)+, (z)) be an arbitrary function localized on Λ+. Then = ZA(βΓίjd(z)d(y)+d(y)_e-βH'Ψ((y)+

Λz))θιF((y)_ ,(z))

i.e. 0,

(2.2)

which is reflection positivity. Clearly, this form of reflection positivity also holds for the classical JV-vector models. Next, we consider the quantum mechanical models and the classical iV-vector models. We let / be a pair of lines between lattice lines cutting A into two disjoint, congruent pieces, A+ and A_. Let 3I7 denote the family of all bounded functions of the spin S. ("algebra of observables" at site;). We define

Phase Transitions in Anisotropic Lattice Spin Systems

249

and Given some £e9I, we define ΘB = θfi by (2.3)

(ΘB)(($)Λ) = B((ΘS)Λ), where =

{SθιJ:jeΛ},

i.e. ΘB is obtained from B by substituting Sθιj for S., all jeΛ. Clearly θ defines an isomorphism from 91 + onto 9Ϊ_ (and conversely). Furthermore we define B = (Bτ)*

(2.4)

to be the complex conjugate (not the adjoint) of B, for arbitrary Be*Ά. Following [9] we study Hamiltonians of the following general form: (2.5)

H = B + θ(B)-ΣCiθ(Q, i

where J5,C 1 ? ...,C k ,... are in 9Ϊ + , (and B = B*,Ci= ±Cf, for all ΐ, so that H is selfadjoint). The following result is a slight variation of Theorem E.I of [9]. Theorem 2.1. (Reflection Positivity). Let F e 2 ί + . Then

where "Tr" means the usual trace in the quantum case and an integral in the classical case. Proof. It clearly suffices to prove that Ύr(e~βHF(ΘF)) ^ 0. By the Trotter product formula, e~βH = lim Gπ,

where

n—• oo

—Γ

(

-(β/n)Be-(β/n)ΘB\ j

e

R i 0_

L n

Thus, Theorem 2.1 is proved if Ύr(GnF(ΘF)) ^ 0, for all n. (2.7) To prove (2.7), note that all elements in 91+ commute with all elements in 91 _. In (2.7) all elements with a θ (which are in 9Ϊ__) can therefore be moved to the right of all elements without a θ (which are in 91+). This shows that Tr(GnF(BF)) is a sum of terms of... the form , ...D Fθ5 ΘD ΘF) = Ύr(D ...DJθφ,... D F)) m

1

m

=

1

m

Ύr(D1...DmF)Ύr(Dι...DmF),

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J. Frόhlich and E. Lieb

with Dί,..., Dm in 21+ . Here we have used the obvious facts that Tr (AB) — Tr (A) Ίx(B\ for AeSΆ+ , J3e2I_, and Ύτ(ΘA) = Tr(A\ for all Ae B).

Q.E.D.

We leave it to the reader to check that the Hamiltonians HΛ of models (1), (3) and (6) are of the form (2.5). See also [9]. Hence Theorem 2.1 proves reflection positivity, inequality (1.39), for these models. However, for the quantum ferromagnet, models (4) and (5) (ferromagnetic case), HΛ is not of the form (2.2) (because of the SfSJ terms), and the proof of Theorem 2.1 breaks down. At present, no useful form of reflection positivity is known for these models. In the sequel, we will assume that inequality (1.36), which follows from reflection positivity (as shown in Section II.B) does hold for the ferromagnetic models, even though we have no proof of it. II.B. Chessboard Estimate Our goal in this subsection is to use reflection positivity to prove inequalities (1.42) and (1.44) (chessboard estimate). We prove a general theorem that includes (1.42) as a special case. Theorem 2.2. (Generalized Holder Inequality). Let 21 be a vector space with antilinear involution J (to be thought of as complex conjugation). Let ωbe a multilinear functional on 21 x 2M,for some integer M > 0, with the properties (C)ω(Ax,...,A2M)

= ω(A29...9A2M9A1)

(cyclicityl and

(θ) The matrix K whose matrix elements K(j are given by K-ij — ω ( ^ i , i '

' JAtM, A.Λ,...,

AjM),

with Alm an arbitrary vector in 21, for all 1= 1, . . . , n , m = 1,...M, is a positive semi-definite nx n matrix, for all n= 1,2,... (Reflection Positivity). Then 2M

(1) \ω(Aί,...,A2M)\S

Y[ω(JApAp...JApA

(chessboard estimate) and (2) \\A\\2M is a semi-norm on 2Ϊ. Proof. 1. A Schwarz Inequality. Let 5£(21x M ) be the vector space over the complex numbers spanned by all elements in 2ί x M . Hypothesis (θ) tells us precisely that ω defines an inner product on i ? ( 2 I x M ) . As a special consequence of the Schwarz inequality for this inner product we have ω(JA2M,...,

JAM+!,

AM

,..., A2M)

+ι

.

(2.8)

Phase Transitions in Anisotropic Lattice Spin Systems

251

2. Proof of Theorem 22 for M = 2. This serves to exhibit the main ideas behind the proof of the general case. By (2.8) and hypothesis (C), I ω(A, B, C, D) I ^ ω(A, B, JB, JΛ)ίl2ω(JD, JC, C, D)1/2 = ω(£, JB, JΛ, A)1/2ω{JC, C, A JD)1/2 ^ ω(B, JB, £, JB)1/4ω(JA, A, JA, A)11* ω{JC, C,JC,

C)ίl4ω(D,JD,D,JD)ί/4

= ω(JA, A, JA, A)1/4ω(JB, £, JB, B)1/4 • ω(JC, C, JC, C)1/4ω(JD, A JD, D)ί/4 which is (1); (2) follows from the multilinearity of ω and (1). 3. The General Case. Since ω is multi-linear and ω(JAj,Aj,..., JAj^Aj) = ω{Aj,JAj,..., Ap JA),

by hypothesis (C), we may assume that (2.9)

ω(JApAp...,JApA)=l, ll2M

for all j = 1,...,2M; (if not, replace ^ by ω{JApAp...,JAvA)~ Ά). We set JAj = Aj+2MJ=l,...i2M. A configuration c is a function on {1,...,2M} with values in {1,...,4M}. Let z = max|ω(^ c ( 1 ) ,^ c ( 2 ) ,...,^ c ( 2 M ) )|, i.e. ( )

forallc.

(2.10)

Lemma, z = 1.

Proof. For c defined by c(2m - 1) =j + 2M, c(2m) = j , m= 1, ...,M, by (2.9). Hence z ^ 1. Thus, it suffices to show z ^ 1. Let c be a configuration for which

Let c(M + 1) = j . Then, by the Schwarz inequality (2.8),

z = I ω(Ad(1),..., •ω ( J ^ 4 ί ( 2 Λ f ) ' 1

Aέ(2M) I >J ^ j , ^ , . . . ,

A~c{2M))

2

^ z > co(JA,i2M),...,JApAp...,AH2M))112, ll2

by (2.10) 112

= z ω(JAdi2M_ίy...,JApAj,...,Aδ{2MyJAδ{2M)) , S z^ω(JAd{2M_

1}

,..., JApApJApAp

...,^(2M_υ)

by hypothesis (C) 1/4

, by (2.8) and (2.10)

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J. Frόhlich and E. Lieb

1 2 r 2 rn ^ z - ~ ω{JApAp...,JApA) ~ , = z 1 - 2 - m ,by(2.9),

by (2.8) and (2.10)

for some m with 2m~ί ^ M > 2 m ~ 2 . Hence z 2 " w ^ 1, i.e. z ^ 1. Q.E.D. To prove Theorem 2.1, (1), let c be given by c(j) =jj = 1,...,2M. By (2.10) and the Lemma, \

(2.11)

\...,A2M)\ 1 and A > 0 to be chosen later. We require nA < 1/40. IVB. The R+

Estimate

We now estimate R+(β,A) for these models. {a') Models (/) and (6 Classical). Let P^*5 be the subset of configurations such that m. = S~ ιSz ^ (1 - δ) exp{ - β T r [ P > d by Jensen's inequality. By symmetry, the term proportional to α vanishes in Tr P%*HA. Furthermore, HΛ^-2\A\{\-δ) whenever P>δ j= 0. Moreover, (4.9) Hence, choosing δ = A/4 = (1 — α)/8, we obtain Λ + ( j M ) ^ exp[j8|/l|(2 - ^ ί j Z ; 1 S Γ ^ ^ ^ ^ ^ ^ l ,

(4.10)

where c(α) oc α — ln(l — α) for α ^ 1, is independent of jS. Thus i/μi

=

RA(β)V\Λ\ = R^A)1^

^ e-W*>(i-«)+c(«),

(4.11)

which tends to 0, as β -> oo. T/zi5 completes the proof of LRO for models (1) and (6, classical) /or /αr^fβ enough β. An estimate on the spontaneous magnetization (moy = S~ί(Szoy, resp. σ(/?) (see Section I.B) as a function of β is given later. (//) Model (2). By definition of model (2) (anharmonic crystal, Section LA), min φ(x,y) = ε0, occurs when x and y have the same direction. Without loss of generality we may assume that there exist some x0 φ 0 such that #*o>*o) = βo 1

(4

12

)

1

But when x and y (the 1-components of x, resp. y) have opposite sign 0(x, j;) ^ ε 0 + α + Λiφt(x) + φ^)),

(4.13)

for some λ > 0; see Section LA. We now choose δ > 0 such that xj - 0 and

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J. Frόhlich and E. Lieb

for all x and y in a ball of radius δ centered at x0. We can do so, since the interaction potential φ is by assumption continuous Hence ZΛ(β) ^

^

where vN(δ) is the volume of a ball of radius δ in MN. Furthermore, for A = α/2 i.e. jRl c (M) = 0 (see (4.4)), S exp[ - 2β(ε0 + α / ^ l J ^ ' Z ^ Γ 1 ,

(4.15)

where This is an immediate consequence of (4.13); (see also inequality (1.49) of Section I.E). Combination of (4.14) and (4.15) yields

Rf{β,aβ) S exp[ - (βφ)\Λ\-](g(β)/vN(δ)rK By definition of the AC model (model (2), Section LA.) there exists some finite β0 such that for all β ^ β0

Obviously g(β) is monotone decreasing in /?, as φχ is positive. Thus there exists a finite constant c such that Hence RΛC(β) = Rf(β,a/2) S exp[ - j3α/4 + ό]\Λ\9

(4.16)

which tends to 0 as β -» oo. Recalling condition (4.1) (resp. Theorem 1.1 of Section I.C and Section I.D, inequality (1.38)) we observe that inequality (4.16) completes the proof of XRO for the AC model for large enough β. (d) Models (3) and (4) (Quantum Heisenberg Models). In order to estimate R+(β, A) we need a lower bound on the partition function ZΛ(β). This is done by comparing it with the partition function of the corresponding spin S Ising model (anisotropy α = 0) by means of the Peierls-Bogolyubov inequality. Lemma 4.1. For models (3) and (4) the partition function satisfies ZΛ^Z'Λ.

(4.17)

Where Z'js the partition function of the spin S Ising model (i.e. α = 0 in (1.3) and (1.5)). Proof. By the Peierls-Bogolyubov inequality,

4δ (ferromagnet),

(4.19)

(1.50) and (1.51) are established for β sufficiently large. For the antiferromagnet, e0 > - 21AI (1 + 1/4S). Thus, provided A > 4δ + S~ * (antiferromagnet),

(4.20)

(1.50) and (1.51) are established for β sufficiently large. The final estimate for the ferromagnet is obtained from (4.6) and (1.51). Choose 2/3 n = (1 + α)/(l - α) < 2/(1 - α). Thus σ < (1 + α)/2 < 1. Choose A = Kβ~ where X is chosen such that σ(^m^onK)-^/ιβ < β-x/2 τ h i s c a n b e d o n e u n j f o r m i y j n S > 1/2. For sufficiently large /?, nJ < 1/40. Furthermore, with this choice of A,R+ > R_. Hence 1

1

1/3

lim ( P ^ ) ^ g exp( - Kβ /2)

(4.21)

which tends to zero as β tends to infinity. Note that there is a /?1/3, instead of a /? dependence in (4.21). This completes the proof of LRO for the quantum ferromagnet, except for the assumption that the chessboard estimate holds. The calculation for the antiferromagnet is more complicated. We have to combine (4.5) with (1.51). As β -+ oo,£ + (jM) 1 / M I -» 0, provided ξ < 1, by (1.51) and (4.5). The problem resides in R_(β,A). This will not go to zero as β -> oo, but for small enough α (depending on S), which we call αc(S), we can make R_(β, /\y/\Λ\ smaller than any given number, say μ. Choose μ such that (4.1) is satisfied. We omit details, but note that ac(S) tends to 1 as S tends to infinity.

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J. Frόhlich and E. Lieb

IV.C. Estimate of the Spontaneous Magnetization Consider the order parameter which satisfies the previously derived inequality 2

σ{β)>δ -6ε

(1.24)

provided < ε/2 and < P Q *> < ε. In the classical models (1) and (6) these inequalities hold for all ε > 0 and δ < 1 if β is large enough. This follows from 3 chessboard estimates applied to (P^ }, and the results of Section IV. Thus σ(β) -> 1 as β -> oo. 6 For the quantum antiferromagnets (3), (6), an estimate on (PQ } can be obtained using chessboard estimates and exponential localization, as before, with the following result: Given ε > 0, δ < 1 and α < 1 there exists an S(ε, δ9 α) < oo such that (1.24) holds as β -+ oo for S > S(s,δ,(x). For the ferromagnet (4), chessboard estimates, if they could be shown to be true, would easily yield σ(β) -> 1 as β -» oo for all S and all a < 1. Without using chessboard estimates, we can a s show that < P Q ^) ""* 0 β ~* °° for all δ < 1 and all α < 1. This is proved by means of the following thermodynamic argument: It is sufficient to show

By the Schwarz inequality and translation invariance

This latter quantity is half the Hz-energy per site. The ground state has the property that S " 2 < S ? S p = l for all i,j. If S" 2 -f» 1 as β-> oo, for | ι - O | = l, then the free energy would not approach the ground state energy as β -> oo. This, it is easy to see by the previous arguments, would be a contradiction. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Mermin, N., Wagner, H.: Phys. Rev. Letters 17, 113 (1966) Mermin, N.: J. Math. Phys. 8, 1061 (1967) Hohenberg, P.: Phys. Rev. 158, 383 (1967) Malyshev, S.: Commun. math. Phys. 40, 75 (1967) Kunz, H., Pfister, C, Vuillermot, P.: J. Phys. A 9, 1673 (1976); Phys. Lett. 54A, 428 (1975) Ginibre, J.: Commun. math. Phys. 14, 205 (1969) Robinson, D.: Commun. math. Phys. 14, 195 (1969) Frόhlich, J., Lieb, E.: Phys. Rev. Letters 38, 440 (1977) Dyson, F., Lieb, E., Simon, B.: Phys. Rev. Letters 37, 120 (1976). The details are in: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. (in press) Frόhlich, J.: Acta Phys. Austriaca, Suppl. 15, 133 (1976) Lieb, E.: New proofs of Long Range Order, Proceedings of the International Conference on the Mathematical Problems in Theoretical Physics, Rome 1977. Lecture notes in physics. BerlinHeidelberg-New York: Springer (in press) Frόhlich, J., Israel, R., Lieb, E., Simon, B.: (papers in preparation) Heilmann, O., Lieb, E.: Lattice models of liquid crystals (in preparation) Lieb, E.: Commun. math. Phys. 31, 327 (1973) Brascamp, H., Lieb, E.: Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In: Functional integration and its applications (ed. A. M. Arthurs), pp. 1-14. Oxford: Clarendon Press 1975

Phase Transitions in Anisotropic Lattice Spin Systems 16. 17. 18. 19. 20.

21. 22. 23.

267

Glimm, J., Jaffe, A., Spencer, T.: Commun. math. Phys. 45, 203 (1975) Anderson, P. W.: Phys. Rev. 83, 1260 (1951). See also the second paper cited in Ref. [9] Osterwalder, K., Schrader, R.: Commun. math. Phys. 31, 83 (1973); 42, 281 (1975) Frόhlich, J.: Helv. Phys. Acta 47, 265 (1974) Seiler, E., Simon, B.: Ann. Phys. (N.Y.) 97, 470 (1976) Frόhlich, J., Simon, B.: Ann. Math 105, 493 (1977) Frohlich, J., Spencer, T.: In: New developments in quantum field theory and statistical mechanics (eds. M. Levy, P.Mitter). New York: Plenum Publ. Corp. 1977; (Theorem 5.2, Lemma 5.3). The basic idea is due to: Frohlich, J., Simon, B., Spencer, T.: Commun. math. Phys. 50, 79 (1976) McBryan, O, Spencer, T.: Commun. math. Phys. 53, 299 (1977) Hepp, K., Lieb, E.H.: Ann. Phys. (N.Y.) 76, 360 (1973), (Theorem 3.15) Peierls, R.: Proc. Cambridge Phil. Soc. 32, 477 (1936) Griffiths, R.: Phys. Rev. 136A, 437 (1964) Dobrushin, R.: Dokl. Akad. Nauk SSR 160, 1046 (1965)

Communicated by J. Glimm Received December 29, 1977

Note Added in Proof

We are informed that an inequality similar to Theorem 2.2 was also proved independently by R. F. Streater and E. B. Davies (unpublished). Their proof is similar to ours.